curve #5603

y² + y = x³ − 128407020085825053

a-invariants: [0, 0, 1, 0, -128407020085825053]
c-invariants: [0, 110943665354152845576]
rank (lower bound): ≥ 12
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 2374324244254292909372143153853800689
discriminant (Δ): -7122972732762878728116429461561402067
Faltings height: 5.7502
naive height: 92.3111
primes of bad reduction: 3, 239, 1619, 84947, 15626293
regulator: 16275866442094612013.956029825023646465606245175506931727547464
submitted by: jesper-petersen
submitted at: 2026-10-23 22:23:16 UTC
last updated: 2026-10-23 22:23:16 UTC

Witness: 12 independent points log in to add more points →

(167591442508583633/49021502464, 68498144677307587391384423/10853752817549312)
(16818426292499202529/18514122262416, -62788236220005759874571517937/79662639327212614464)
(133797018498546523105/218872969454736, 1024116241376895534346556130481/3238084628890490270016)
(1408403430777361241415486097/1304824635211877750916, 50084393900609849124787898303683789164035/47133340572645370851564555004536)
(2678688243069702077681/120833077608100, -138637493495646806306172577128421/1328246730630054521000)
(1616519958166617638247924553/2251025143001842458384, -52531432655501805750755910091686854124955/106799819541586772186069733961152)
(213808451506392917138300403425/404176687312887746076304, -35998232686840423890987897710712410844796111/256954891822680574688894274665561408)
(38718984383927267264525705201/25853125866924914469604, -7471753008450421349255735151464052995868091/4156900131902221759668098220311192)
(9571536975706626418088577349697/158473008833206054385476, 29612317387915929656574251988551235477417809605/63085994760113718765901101011585224)
(8319138781841066623182843799556929/6751403194051637086185864900, -732281181603014307677731794569366080555721183887283/554742024693465319001201030890083620643000)
(123957364821090108914569979350056545/54496001427702983588975826916, -43403642982339614468140916600319759545734013535483163/12721752919152821185486645129809400427151464)
(5826248573341479019049249671620540999381841/9335447416337899143653713007988294564, 9659316261682931832377987414374520685401903929186853234081879179/28523492818144252381541675178543159755137135736304409512)

Commentary

Rank-12 Mordell curve constructed using the Burkhardt quartic family, with parameters (u,v,w)=(-2,-9,-12) and specialization t=407/102.

last edited by jesper-petersen at 2026-10-23 22:23:17 UTC · history